nLab Milnor slide trick

This page will probably have to be renamed something like “fiber bundles are fibrations” once I remember how the trick works in detail.

Zoran Škoda: But there is much older and more general theorem of Hurewitz: if one has a map $p:E\to B$ and a numerable covering of $B$ such that the restrictions ${p}^{-1}\left(U\right)\to U$ for every $U$ in the covering is a Hurewicz fibration then $p$ is also a Hurewicz fibration. But the proof is pretty complicated. For example George Whitehead’s Elements of homotopy theory is omitting it (page 33) and Postnikov is proving it (using the equivalent “soft” homotopy lifting property).

Todd Trimble: Yes, I am aware of it. You can find a proof in Spanier if you’re interested. I’ll have to check whether the Milnor trick (once I remember all of it) generalizes to Hurewicz’s theorem.

Stephan: I wonder if this trick moreover generalizes (in a homotopy theoretic sense) to categories other that $Top$; for example to the classical model structure on $\mathrm{Cat}$?

Let $\pi :E\to B$ be a principal $G$-fiber bundle which has a numerable cover (this condition obtains if for example $B$ is paracompact). Suppose given a commutative diagram in Top:

$\begin{array}{ccc}X& \stackrel{f}{\to }& E\\ {i}_{0}↓& & ↓\pi \\ X×I& \underset{\varphi }{\to }B\end{array}$\array{ X & \overset{f}{\to} & E \\ i_0 \downarrow & & \downarrow \pi \\ X \times I & \underset{\phi}{\to} B }

where ${i}_{0}$ is the composite inclusion $X\cong X×\left\{0\right\}↪X×I$. We are trying to show that $\varphi$ lifts through $\pi$.

As I recall, the trick proceeds by considering the bundle

$\begin{array}{c}\left({\varphi }_{0}{\right)}^{*}E×\left(-\infty ,0\right]\cup {\varphi }^{*}E\cup \left({\varphi }_{1}{\right)}^{*}E×\left[1,\infty \right)\\ ↓\\ \left(X×\left(-\infty ,0\right]\right)\cup \left(X×I\right)\cup \left(X×\left[1,\infty \right)\end{array}$\array{ (\phi_0)^*E \times (-\infty, 0] \cup \phi^* E \cup (\phi_1)^* E \times [1, \infty) \\ \downarrow \\ (X \times (-\infty, 0]) \cup (X \times I) \cup (X \times [1, \infty) }

where the base is $X×ℝ$ and ${\varphi }_{t}$ is the restriction of $\varphi$ along $X×\left\{t\right\}↪X×I$, and then one constructs a bundle lifting? of the homeomorphism

$X×ℝ\to X×ℝ:\left(x,t\right)↦\left(x,t+1\right)$X \times \mathbb{R} \to X \times \mathbb{R}: (x, t) \mapsto (x, t + 1)

This bundle lifting is “the slide”. Now the bundle is trivial over $X×\left[-1,0\right]$ (see below), so it has a section, and one transports this section along the slide to give a section $\sigma$ over the part

$\begin{array}{c}{\varphi }^{*}E\\ ↓\\ X×\left[0,1\right]\end{array}$\array{ \phi^* E\\ \downarrow \\ X \times [0, 1] }

and then the composition

$X×I\stackrel{\sigma }{\to }{\varphi }^{*}E\to E$X \times I \overset{\sigma}{\to} \phi^* E \to E

gives the desired homotopy lifting?.

To see that the bundle restricted over $X×\left(-\infty ,0\right]$ is trivial, we just need to check that $\left({\varphi }_{0}{\right)}^{*}E$ is trivial over $X$. However, by the original commutative square, ${\varphi }_{0}$ equals the composite

$X\stackrel{f}{\to }E\stackrel{\pi }{\to }B$X \overset{f}{\to} E \overset{\pi}{\to} B

and already ${\pi }^{*}E$ is trivial (over $E$) essentially because $\pi$ is a $G$-torsor: there is a bundle isomorphism

${\pi }^{*}E\cong E{×}_{B}E\to E×G$\pi^* E \cong E \times_B E \to E \times G

over $E$.

The construction of the slide is where transfinite composition comes in. The details are at this moment a little hazy, but the rough idea is to construct a partition of unity ${\rho }_{\alpha }$ subordinate to the pulled back (locally finite) numerable cover ${U}_{\alpha }$ of $X×I$. One is supposed to well-order the $\alpha$, and then transfinitely compose a bunch of mini-slides over $\left(x,t\right)↦\left(x,t+{\rho }_{\alpha }\left(x\right)\right)$. The transfinite composition is well-defined on the fiber over any $x$ because the arrows in the composite are non-identity only for those ${U}_{\alpha }$ which contain $x$, and there are only finitely many of these by local finiteness.

To be continued.